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Resonant scattering properties close to a p-wave Feshbach resonance
Frédéric Chevy, E.G.M. Van Kempen, Thomas Bourdel, J. Zhang, L.
Khaykovich, M. Teichmann, L. Tarruell, S.J.J.M.F. Kokkelmans, C. Salomon
To cite this version:
Frédéric Chevy, E.G.M. Van Kempen, Thomas Bourdel, J. Zhang, L. Khaykovich, et al.. Resonant scattering properties close to a p-wave Feshbach resonance. 2005. �hal-00003592v2�
ccsd-00003592, version 2 - 29 Mar 2005
F. Chevy1, E. G. M. van Kempen2, T. Bourdel1, J. Zhang3, L. Khaykovich4, M. Teichmann1, L. Tarruell1, S. J. J. M. F. Kokkelmans2, and C. Salomon1
1 Laboratoire Kastler-Brossel, ENS, 24 rue Lhomond, 75005 Paris
2 Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands
3 SKLQOQOD, Institute of Opto-Electronics, Shanxi University, Taiyuan 030006, P.R. China and
4 Department of Physics, Bar Ilan University, Ramat Gan 52900, Israel.
(Dated: March 29, 2005)
We present a semi-analytical treatment of both the elastic and inelastic collisional properties near a p-wave Feshbach resonance. Our model is based on a simple three channel system that reproduces more elaborate coupled-channel calculations. We stress the main differences between s-wave and p-wave scattering. We show in particular that, for elastic and inelastic scattering close to a p-wave Feshbach resonance, resonant processes dominate over the low-energy behaviour.
PACS numbers: 03.75.Ss, 05.30.Fk, 32.80.Pj, 34.50.-s
INTRODUCTION
The observation of molecular gaseous Bose-Einstein condensates (BEC) and the subsequent experimental study of the BEC-BCS crossover [1, 2, 3, 4, 5] were made possible by the possibility of tuning interatomic inter- actions using a magnetic field (the so-called Feshbach resonances). Although all these experiments were based on s-wave interatomic interactions, it is known from con- densed matter physics that superfluidity of fermionic sys- tems can also arise through higher order partial waves.
The most famous examples of this non-conventionnal su- perfluidity are3He [6], for which the Cooper pairs spawn from p-wave interactions, and high-Tcsuperconductivity, in which pairs are known to possess d-wave symmetry [7].
Recent interest in p-wave interactions in cold atom gases stemmed from these possibilities and resulted in the ob- servation of p-wave Feshbach resonances in 40K [8] and
6Li [9, 10], as well as theoretical studies on the super- fluidity of cold atom interacting through p-wave pairing [11, 12].
The present paper is devoted to the study of p-wave interactions close to a Feshbach resonance and it derives some results presented in [9]. In a first part, we present the model we use to describe both elastic and inelastic processes that are discussed in the second part. We stress in particular the main qualitative differences between p- wave and s-wave physics and show that contrarily to the case of s-wave that is dominated by low energy physics, p-wave scattering is dominated by a resonance peak as- sociated to the quasi bound molecular state. Finally, we compare our analytical results to numerical coupled- channel calculations.
MODEL FOR P-WAVE INTERACTIONS
We consider the scattering of two identical particles of mass m. As usual when treating a two-body problem,
we work in the center of mass frame and only consider the motion of a fictitious particle of massm/2 interact- ing with a static potential. In order to study the p-wave Feshbach resonance, we use a model based on the separa- tion of open and closed channels. In this framework, the Feshbach resonance arises in an open channel as a result of the coupling with a closed channel [13]. At resonance, scattering properties are dominated by resonant effects and we can neglect all “background” scattering (ie. we assume there is no scattering far from resonance).
1. We restrict ourselves to a 3-channel system, la- belled 1,2 and 3 which correspond to the different two-body spin configurations (Fig. 1). Channels
|1i and |2i are open channels. We focus on the situation were atoms are prepared initially in state
|1i. Atoms may be transferred to state|2iafter an inelastic process. Channel |3i is closed and hosts the bound state leading to the Feshbach resonance.
Let us consider for instance the case of6Li atoms prepared in a mixture of |F = 1/2, mF = 1/2i and |F′ = 1/2, m′F = −1/2i. In this system, the only two-body decay channel is associated with the flipping of a m′F = −1/2 atom to m′F = 1/2.
If we denote by (mF, m′F) the symmetrized lin- ear combination of the states |F = 1/2, mFi and
|F′ = 1/2, m′Fi, then |1i= (1/2,−1/2) and |2i = (1/2,1/2).
2. The Feshbach resonances studied here are all lo- cated at values of the magnetic field where the Zeeman splitting is much larger than the hyperfine structure. In first approximation we can therefore assume that the internal states are described by uncoupled electronic and nuclear spin states. In the absence of any dipolar or hyperfine coupling between the electronic singlet and triplet mani- folds, we assume we have no direct interaction in channels 1 and 2 so that the eigenstates are plane waves characterized by their relative wave-vectork
2 and their energyE1(k) = ¯h2k2/m(channel 1) and
E2(k) =−∆ + ¯h2k2/m (channel 2). ∆>0 is the energy released in an inelastic process leading from 1 to 2. ∆ can be considered as independent of the magnetic field and is assumed to be much larger than any other energy scales (in the case relevant to our experiments, ∆/h∼80 MHz is the hyperfine splitting of6Li at high field).
3. In channel 3, we consider only a p-wave bound state nesting at an energyδquasi-resonant with channel 1. In the case of 6Li atoms in the F = 1/2 hy- perfine state, δ = 2µB(B−B0), where B is the magnetic field andB0is the position of the “bare”
Feshbach resonance. If the projection of the an- gular momentum (in unit of ¯h) is denoted by mu
for a quantization axis chosen along some vectoru, the eigenfunctions associated to this bound state can be written as g(r)Y1mu(θ, φ), where (r, θ, φ) is the set of polar coordinates and the Ylm are the spherical harmonics.
4. The coupling Vb between the various channels af- fects only the spin degrees of freedom. Therefore the orbital angular momentum is conserved during the scattering process and we restrict our analy- sis to the p-wave manifold. This is incontrast to the situation in heavy alkalis where incoming par- ticles in s-wave can be coupled to molecular states of higher orbital angular momentum [14, 15].
We assume also that the only non-vanishing ma- trix elements are between the closed and the open channels (ie. h1,2|Vb|3iandh3|Vb|1,2i).
Let us denote a state of the system by |α, χi, where α ∈ {1,2,3} and χ describe respectively the internal (spin) and the orbital degrees of freedom. According to assumption (4), the matrix elementhα, χ|Vb|α′, χ′iis sim- ply given by:
hα, χ|Vb|α′, χ′i=hχ|χ′ihα|Vb|α′i, (1) and is therefore simply proportional to the overlaphχ|χ′i between the external states.
Let us now particularize to the case where α ∈ {1,2}, and |χi = |ki is associated to a plane wave of relative momentum ¯hk. According to hypothesis (4), this state is coupled only to the closed channel
|3i. Moreover, using the well known formula eikz = P
lilp
4π(2l+ 1)jl(kr)Yl0(θ, φ), where the jl are the spherical Bessel functions, we see that|χiis coupled only to the state |α′ = 3,k = 0i describing the pair in the bound state |α′ = 3i with zero angular momentum in thekdirection. The matrix element then reads
D d
|1>
|2>
|3>
V^
FIG. 1: The p-wave model: we consider three internal states, labelled|1i,|2iand|3i. States|1iand|2iare two open chan- nels corresponding respectively to the incoming and decay channels. The released energy in an inelastic collision bring- ing atom initially in |1i to |2i is denoted ∆. State |3i is a closed channel that possesses a p-wave bound state of energy δnearly resonant with state|1i. Finally, we assume that these three channels interact through a potentialVb acting only on the internal states and coupling the two open channels to the closed channels.
hα,k|Vb|3, mki=iδmk,0
r12π
L3 hα|Vb|3i Z
g∗(r)j1(kr)r2dr, (2) whereL3 is a quantization volume. Since for smallkwe havej1(kr) ∼ kr/3, the matrix element hα,k|Vb|3, mki takes the general form
hα,k|Vb|3, mki=δmk,0
kFα(k)
√L3 , (3) whereFα(k) has a finite (in general non zero) limit when kgoes to zero.
Later on, we shall also need the coupling between|α,ki and|3, mk′ = 0i(that will be denoted by|3,0k′i), where the momentum k and the direction of quantization k′ are no longer parallel. The calculation presented above yields readily
hα,k|Vb|3,0k′i= kFα(k)
√L3 h0k|0k′i= kFα(k)
√L3 cos(k,dk′), (4) where (k,dk′) is the angle between kandk′ [16].
T-MATRIX
From general quantum theory, it is known that the scattering properties of a system are given by the so- called T-matrix Tb. It can be shown in particular that
|3,0k >1
|a,k> |a1,k1> |a2,k2> |an,kn> |a ,k >
|3,0k> |3,0k >
T n
aa (k,k )=
S S S
nap=1,2 kp p=1
n
¢
¢
¢ ¢
FIG. 2: Diagrammatic expansion of theT-matrix. The full lines (resp. dashed) represent free atoms (resp. molecules). |α,ki is the scattering state of the two particles, in the internal stateα= 1,2. |3,0kirepresents the state of a p-wave molecule with orbital angular momentum component zero onkdirection.
Tb is given by the following expansion in power of the coupling potential
Tb(E) =Vb +VbGb0(E)bV +VbGb0(E)VbGb0(E)Vb+..., (5) whereGb0(E) = 1/(E−Hb0) andHb0=Hb−Vb is the free hamiltonian of the system.
Let us consider |α,ki and |α′,k′i, two states of the open channels and let us set Tαα′(k,k′, E) = hα,k|Tb(E)|α′,k′i. According to formula (5), this matrix element is the sum of terms that can be represented by the diagram of Fig. 2 and we get after a straightforward calculation
Tαα′(k,k′, E) = kk′
L3Fα(k)Fα∗′(k′) X∞
n=0
RnΛ(E)nG(m)0 (E)n+1.
Here G(m)0 (E) = 1/(E−δ) is the free propagator for the molecule, Λ = Λ1+ Λ2 with
Λα(E) =
Z q4dq (2π)3
|Fα(q)|2
E−Eα(q) (6) results of the integration on the loops, and finally
Rn= Z
d2Ω1...d2Ωncos(k,dk1) cos(kd1,k2)...cos(kdn,k′), where Ωp is the solid angle associated tokp, arises from the pair breaking vertices |3,0kii → |αi+1,ki+1i. This last integral can be calculated recursively and we get Rn = (4π/3)ncos(k,dk′), that is for theT-matrix
Tαα′(k,k′, E) = 1 L3
kk′Fα(k)Fα∗′(k′) E−δ−Σ1−Σ2
cos(k,dk′), with Σα= 4πΛα/3.
This expression can be further simplified since, accord- ing to Eqn. (2), the width ofFα(q) is of the order of 1/Re, whereReis the characteristic size of the resonant bound state. In the low temperature limit, we can therefore expand Σαwith the small parameterkRe.
From Eqn. (2), we see that replacingFα(q) by its value atq= 0 leads to aq2divergence. This divergence can be regularized by the use of counter terms in the integral, namely by writing that
Σα(E) = Z
|Fα(q)|2
q4
E−Eα(q)+mq2
¯
h2 +m2
¯
h4(E−Eα(0)) dq
6π2
− Z
|Fα(q)|2mq2
¯ h2
dq
6π2−(E−Eα(0)) Z
|Fα(q)|2m2
¯ h4
dq 6π2, where we have assumed that Fα was decreasing fast enough at largeq to ensure the convergence of the in- tegrals. |Fα(q)|2 can now be safely replaced by λα =
|Fα(0)|2 in the first integral and we finally get
Σα=−iλα
6π m
¯ h2
m
¯
h2(E−Eα(0)) 3/2
−δ0,α−ηα(E−Eα(0)), with
δ0(α) = Z
|Fα(q)|2mq2
¯ h2
dq 6π2 ηα =
Z
|Fα(q)|2m2
¯ h4
dq 6π2.
If we assume that the release energy ∆ is much larger thanE and if we setδ0=δ(1)0 +δ(2)0 andη=η1+η2, we get for theT-matrix
Tαα′(k,k′, E)≃ 1 L3
kk′Fα(0)Fα∗′(0) cos(k,dk′)/(1 +η) E−˜δ+i¯hγ(E)/2 .
(7) with
¯
hγ(E) = m
¯ h2
5/2 λ2∆3/2+λ1E3/2 3π(1 +η) δ˜ = (δ−δ0)
1 +η .
We note that this expression for the T-matrix is con- sistent with the general theory of multichannel scattering
4 resonances [13] , where resonantly enhanced transitions
to other channels are readily included. In a similar con- text of two open channels and a Feshbach resonance, a recent experiment was analyzed [15], that involved the decay of a molecular state formed from a Bose-Einstein condensate.
S-WAVE VS P-WAVE
This section is devoted to the discussion of the expres- sion found for the T-matrix. In addition to the scat- tering cross section, the study of the T-matrix yields important informations on the structure of the dressed molecular state underlying the Feshbach resonance and we will demonstrate important qualitative differences be- tween the behaviours of p-wave and s-wave resonances.
Molecular state. The binding energy Eb of the molecule is given by the pole ofT. In the limitδ∼δ0, it is therefore given by:
Eb= ˜δ−i¯hγ(˜δ)/2,
We see that the real part of Eb (the “physical” bind- ing energy) is ∼δ˜and therefore scales linearly with the detuningδ−δ0. This scaling is very different from what happens for s-wave processes where we expect a (δ−δ0)2 behavior. This difference is in practice very important:
indeed, the molecules can be trapped after their forma- tion only if their binding energy is smaller than the trap depth. The scaling we get for the p-wave molecules means that the binding energy increases much faster when we increase the detuning than what we obtain for s-wave molecules (this feature was already pointed out in [11]).
Hence, p-wave molecules must be looked for only in the close vicinity of the Feshbach resonance – for instance, for η ≪ 1 (relevant for 6Li, as we show below) and a trap depth of 100µK, the maximum detuning at which molecules can be trapped is≃0.1 G.
This asymptotic behavior of the binding energy is closely related to the internal structure of the molecule.
Indeed, the molecular wave function|ψm(B)ican be writ- ten as a sum |openi+|closedi of its projections on the closed and open channels, that correspond respectively to short and long range molecular states. If we neglect decay processes by settingλ2= 0, we can define the mag- netic moment of the molecule (relative to that of the free atom pair) by
∆µeff(B) =−∂Eb
∂B =−∂δ˜
∂B,
that is, in the case of6Li whereδ= 2µB(B−B0),
∆µeff(B) =−2µB
1 +η. (8)
On the other hand, we can also write Eb = hψm(B)|Hb(B)|ψm(B)i. Since in the absence of any de- cay channel, the molecular state is the ground state of the two-body system, we can write using the Hellman- Feynman relation
∆µeff =−∂Eb
∂B =−hψm(B)|∂Hb(B)
∂B |ψm(B)i. In our model, the only term of the hamiltonian depend- ing on the magnetic field is the energyδ= 2µB(B−B0) of the bare molecular state in the closed channel and we finally have
∆µeff =−2µBhclosed|closedi. (9) If we compare Eqn. (8) and (9), we see that the proba- bilityPclosed=hclosed|closedito be in the closed channel is given by
Pclosed= 1/(1 +η).
In other words, unless η =∞, there is always a finite fraction of the wave-function in the tightly bound state.
In practice, we will see that in the case of 6Li, η ≪ 1.
This means that the molecular states that are nucleated close to a Feshbach resonance are essentially short range molecules. On the contrary, for s-wave molecules,Eb ∝ (δ−δ0)2leads to ∆µeff =−2µBhclosed|closedi ∝(δ−δ0).
This scaling leads to a zero probability of occupying the bare molecular state near a s-wave resonance.
We can illustrate this different behaviours in the sim- plified picture of Fig. 3. For small detunings around threshold, the p-wave potential barrier provides a large forbidden region, that confines the bound state behind this barrier. The bound state wavefunction decays ex- ponentially inside the barrier and the tunneling remains nearly negligible. Since there is no significant difference for the shape of the p-wave bound state for small posi- tive and negative detunings, the linear dependence of the closed channel with magnetic field will be conserved for the bound state, and therefore the binding energy will also linearly approach the threshold. We note that the linear dependence close to threshold can also be found from the general Breit-Wigner expression for a resonance, in combination with the p-wave threshold behavior of the phase shift [13].
The imaginary part ofEb corresponds to the lifetime of the molecule. In the case of s-wave molecules for which two-body decay is forbidden [20, 21], the only source of instability is the coupling with the continuum of the in- coming channel that leads to a spontaneous decay when
˜δ > 0 (see Fig. 3.a). By contrast, we get a finite life- time in p-wave even at ˜δ <0: due to dipolar relaxation
(a)
(b)
FIG. 3: Effect of the centrifugal barrier on the bound state in p-wave Feshbach resonances. (a) Case of a s-wave scattering:
the bare molecular state goes fromδ <0 (full line) toδ >0 (dashed line). In the process, the molecular state becomes unstable and the wave function becomes unbounded. (b) In the case of p-wave bound state, the presence of the centrifugal barrier smoothes the transition fromδ >0 toδ <0. Even for δ >0, the wave-function stays located close to the bottom of the well.
between its constituents, the molecule can indeed spon- taneously decay towards state|2i. For ˜δ∼0, the decay rateγ0associated to this process is given by:
γ0=γ(0) = λ2
1 +η m 3π¯h3
m∆
¯ h2
3/2
.
Elastic scattering. The scattering amplitude f(k,k′) for atoms colliding in the channel 1 can be extracted from theT-matrix using the relation
f(k,k′) =−mL3
4π¯h2T11(k,k′, E= ¯h2k2/m), that is
f(k,k′) =−mλ1
4π¯h2
k2cos(k,dk′)/(1 +η)
¯
h2k2/m−δ˜−i¯hγ/2
! . The cos(k,dk′) dependence is characteristic of p-wave processes and, once again, this expression shows dramatic
differences with the s-wave behavior. First, at low k, f(k,k′) vanishes like k2. If we introduce the so-called scattering volumeVs[13] defined by
f(k,k′) =−Vsk2, then we have
Vs= −mλ1
4π¯h2(δ−δ0+i(1 +η)¯hγ0/2) ∼ −mλ1
2π¯h2(δ−δ0), if we neglect the spontaneous decay of the molecule. We see that in this approximation, the binding energyEb of the molecule is given by
Eb=− mλ1
2π¯h2(1 +η) 1 Vs.
In s-wave processes, the binding energy and the scat- tering length are related through the universal formula Eb = −¯h2/ma2. This relationship is of great impor- tance since it allows to describe both scattering prop- erties and the molecular state with the sole scattering length, without having to care with any other detail of the interatomic potential. In the case of p-wave, we see that no such universal relation exists between the scat- tering volume andEb, a consequence of the fact that we have to deal with short range molecular states, even at resonance. We therefore need two independent parame- ters to describe both the bound states and the scattering properties.
In the general case, the elastic cross-sectionσel is pro- portional to|f|2. According to our calculation, we can putσel under the general form
σel(E) = CE2
(E−δ)˜2+ ¯h2γ2/4, (10) whereE = ¯h2k2/m is the kinetic energy of the relative motion andCis a constant depending on the microscopic details of the system. Noticeably, the energy dependence of the cross-section exhibits a resonant behavior atE= ˜δ as well as a plateau whenE→ ∞, two features that were observed in the numerical coupled channel calculations presented in [17]. Once again, this leads to physical pro- cesses very different from what is expected in s-wave scat- tering. Indeed, we know that in s-wave, we havef ∼ −a, as long aska≪1. Sinceais in general non zero, the low energy behavior gives a non negligible contribution to the scattering processes. By contrast, we have just seen that in the case of p-wave, the low energy contribution was vanishingly small (σel ∼E2) so that the scattering will be dominated by the resonant peakE∼δ.˜
Inelastic scattering. For two particules colliding in channel 1 with an energy E = ¯h2k2/m, the
6
0 1 2 3 4 5 6 7
1E-4 0.01 1 100 10000
Energy [µK]
s [10 cm ]
-132 el
FIG. 4: Energy dependence of the elastic cross section. Dots:
numerical closed channel calculation. The left peak corre- sponds toml= 0 and the right peak toml=±1. Full line:
Fits using Eqn. (11)
probability to decay to channel 2 is proportional to ρ2(k′)|T12(k,k′, E)|2 where ρ2 is the density of state in channel 2. Since k′ is given by the energy conservation condition ¯h2k′2/m−∆ = E, and in practice ∆ ≫ E, we see that k′ ∼q
m∆/¯h2 is therefore a constant. Us- ing this approximation, we can write the 2-body loss rate g2(E) for particles of energyE as:
g2(E) = DE
(E−˜δ)2+ ¯h2γ2/4,
whereD is a constant encapsulating the microscopic de- tails of the potential.
COMPARISON WITH COUPLED-CHANNEL CALCULATION
The quantities such asC, D, γ0 etc. that were intro- duced in the previous section were only phenomenological parameters to which we need to attribute some value to be able to perform any comparison with the experiment.
These data are provided by ab initio numerical calcula- tions using the coupled channel scheme described in [18].
The result of this calculation for the elastic scattering cross-section is presented in Fig. 4. The most striking feature of this figure is that it displays two peaks instead of one, as predicted by Eqn. (10). This difference can be easily understood by noting that the dipolar inter- action that couples the molecular state to the outgoing channel provides a “spin-orbit” coupling that modifies the relative orbital angular momentum of the pair [17].
In other word, each resonance corresponds to a different value of the relative angular momentumml(theml= +1
andml =−1 resonances are superimposed because the frequency shift induced by the dipolar coupling is pro- portional tom2l, as noted in [17]).
As the spin-orbit coupling is not included in our sim- plified three-level model, we take the multiple peak struc- ture into account by fitting the data of Fig. (4) using a sum of three laws (10) with a different set of phenomeno- logical parameters for each value of the angular momen- tum:
σel(E) = X+1
ml=−1
CmlE2
(E−δ˜ml)2+ ¯h2γm2l/4, (11) where ˜δmlis related to the magnetic field through ˜δml = µ(B −BF,ml) and γml = γ0,ml +amlE3/2. Using this law, we obtain a perfect fit to the elastic as well as inelas- tic data obtained from the coupled channel calculations.
The values obtained for the different phenomenological parameters are presented on Table I.
From these data, we see first that the “elastic” proper- ties are independent ofml. This comes from the fact that the elastic scattering is mainly a consequence of the hy- perfine coupling that does not act on the center of mass motion of the atoms. However, we see that both the in- elastic collision rate constantDand the molecule lifetime γ0 exhibit large variations with the relative angular mo- mentum [22]. First, the spontaneous decay rateγ0 of a molecule inml= 0,+1 is always larger than ∼102s−1, which corresponds to a maximum lifetime of about 10 ms.
Second, we observe a strong reduction of the losses in the ml=−1 channel, in which no significant spontaneous de- cay could be found. An estimate ofγ0 can nevertheless be obtained by noting that, since the elastic parameters are independent ofml, the ratioD/γ0should not depend onml(this can be checked by comparing the ratiosD/γ0
forml= 0 andml= +1 in the (-1/2,-1/2) channel). Us- ing this assumption we find thatγ0∼4×10−3s−1 both in (1/2,-1/2) and (-1/2,-1/2). The reason for this strong increase of the lifetime of the molecules in ml = −1 is probably due to the fact that due to angular momentum conservation the outgoing pair is expected to occupy a state withl= 3 after an inelastic process. Indeed, if we start in a two-body state (mF, m′F) and if the dipolar relaxation flips the spinm′F, then the atom pair ends up in a state (mF, m′F+ 1). This increase of the total spin of the pair must be compensated by a decrease of the rel- ative angular momentum. Therefore, if the molecule was associated to a relative angular momentumml, it should end up withml−1. In the case ofml=−1, this means that the final value of the relative angular momentum is ml = −2, ie. l ≥ 2. But, according to selection rules associated to spin-spin coupling, the dipolar interaction can only changel by 0 or 2. Therefore, starting from a p-wave (l = 1) compound, this can only lead to l = 3.
Let us now assume that the coupling between the molec-
channel (1/2,-1/2) (-1/2,-1/2)
ml -1 0 1 -1 0 1
C(10−13cm2) 0.22 0.22 0.22 0.87 0.88 0.87
D(10−13cm2µK/s) 0.00002 0.59 0.56 3×10−5 1.54 5.72
γ0(s−1) <10−2 110 110 <1 220 830
a(µK−1/2) 0.0017 0.0017 0.0017 0.0024 0.0024 0.0024
η 0.22 0.22 0.22 0.23 0.23 0.23
δBF(G) -0.0036 0 -0.0036 -0.012 0 -0.012
TABLE I: Values of the phenomenological parameters obtained after a fit to the coupled channel calculations data of Fig. 4.
δBF is the shift between theml=±1 andml= 0 resonances.
ular state and the outgoing channel is still proportional to the overlap between the two states (see Eqn. 1), even in the presence of a dipolar coupling: the argument above indicates that the ratioρ=γ0,ml=−1/γ0,m′l6=−1 between the decay rate of molecules in ml = −1 and the one of molecules inm′l6= 1 is then of the order of
ρ∼
Rg∗(r)j3(kr)r2dr Rg∗(r)j1(kr)r2dr
2
,
where k = q
m∆/¯h2 is the relative momentum of the atoms after the decay. For lithium, we have Re ∼ 3 ˚A [23] which yieldskRe∼7×10−2. This permits to approx- imate the spherical Bessel functionjlby their expansion at lowk, jl(kr)∼(kr)l, that is
ρ∼k4
R g∗(r)r5dr R g∗(r)r3dr
2
∼(kRe)4,
With the numerical value obtained forkRe, we getρ∼ 2×10−5, which is, qualitatively, in agreement with the numerical coupled channels calculations presented above.
TEMPERATURE AVERAGING
In realistic conditions, the two body loss rateG2needs to be averaged over the thermal distribution of atoms. G2
is therefore simply given by
G2(E) =
r π 4(kBT)3
Z ∞ 0
g2(E)e−E/kBTE1/2dE.
The evolution ofG2vs detuning is displayed in Fig. 5 and shows a strongly asymmetric profile that was already noticed in previous theoretical and experimental papers [8, 10].
This feature can readily be explained by noting that in situations relevant to experiments, γ0 is small with respect to temperature. We can therefore replaceg2 by
a sum of Dirac functions centered on δ0,ml. When the δ0,ml are positive,G2 takes the simplified form
G2= 4√ πX
ml
Dml
¯
hγml(˜δml)
˜δml
kBT
!3/2
e−˜δml/kBT. Moreover, if we neglect the lift of degeneracy due to the dipolar interaction coupling and we assume all the
˜δml to be equal to some ˜δ, we get
G2= 4√ π
D¯
¯ h¯γ(˜δ)
˜δ kBT
!3/2
e−δ/k˜ BT, (12) with ¯D=P
mlDml and ¯D/¯γ=P
mlDml/γml [19].
For ˜δ <0, and |δ˜| ≫kBT, we can replace the denom- inator ofg2 by ˜δ2 and we get the asymptotic form for G2
G2=3
2kBTX
ml
Dml
˜δm2l
. (13)
Let us now comment the two equations (12) and (13).
1. We note that the maximum value ofG2is obtained for ˜δ/kBT = 3/2. It means that when tuning the magnetic field (ie., ˜δ), the maximum losses are not obtained at the resonance ˜δ = 0, but at a higher field, corresponding to ˜δ = 3kBT /2. For a typical experimental temperatureT = 10 µK, this corre- sponds to a shift of about 0.1 G.
2. Similarly, the width of G2(˜δ) scales like kBT. Ex- pressed in term of magnetic field, this corresponds to ∼ 0.1 G for T = 10µK. This width is a conse- quence of the resonance nature of the scattering in p-wave processes. As seen earlier, both elastic and inelastic collisions are more favorable when the rel- ative energy E = ˜δ. When ˜δ < 0, the resonance conditions cannot be fulfilled, since there are no state in the incoming channel with negative energy.
